INTERNATIONAL MATHEMATICS OLYMPIAD(IMO)

IMO Course Syllabus

1. Algebra

• Objective: To build strong foundational skills in algebra and problem-solving techniques.


• Topics:
  
  
  
  • Polynomials: Roots and factors, theorems of polynomials, Vieta’s relations, and symmetric polynomials.
  
  
  • Linear Algebra: Matrices, determinants, eigenvalues, and eigenvectors.
  
  
  • Equations: Solving algebraic equations, systems of equations, inequalities (AM-GM, Cauchy-Schwarz).
  
  
  • Functional Equations: Basic functional equations and methods of solving them.
  
  
  
  

2. Combinatorics

• Objective: To enhance the ability to count and organize mathematical objects in various settings.


• Topics:
  
  
  
  • Permutations and Combinations: Basic counting principles, binomial theorem, inclusion-exclusion principle.
  
  
  • Pigeonhole Principle: Applications and problem-solving techniques.
  
  
  • Graph Theory: Basic concepts in graphs, coloring, connectivity, and graph invariants.
  
  
  • Recursion and Recurrence Relations: Solving linear and non-linear recurrences.
  
  
  
  

3. Geometry

• Objective: To develop geometric reasoning and visualization skills.


• Topics:
  
  
  
  • Euclidean Geometry: Properties of triangles, circles, quadrilaterals, and polygons.
  
  
  • Congruence and Similarity: Theorems on angles, parallel lines, and transformations.
  
  
  • Geometric Inequalities: Applications of inequalities in geometry (e.g., triangle inequalities, Euler’s inequality).
  
  
  • Coordinate Geometry: Lines, curves, and distance formula, area of triangles, and conic sections.
  
  
  • Geometric Constructions: Methods of constructing geometrical shapes using classical methods.
  
  
  
  

4. Number Theory

• Objective: To develop techniques for solving problems involving integers and divisibility.


• Topics:
  
  
  
  • Divisibility: Prime numbers, greatest common divisors, and least common multiples.
  
  
  • Modular Arithmetic: Solving congruences and using properties of moduli.
  
  
  • Diophantine Equations: Solving integer solutions to equations.
  
  
  • Prime Factorization and Sieve Methods: Euler’s Totient function, Fermat's Little Theorem, and the Sieve of Eratosthenes.
  
  
  
  

5. Mathematical Induction and Proof Techniques

• Objective: To develop logical reasoning and proof-writing abilities.


• Topics:
  
  
  
  • Mathematical Induction: Strong induction, weak induction, and applications in combinatorics and number theory.
  
  
  • Proof Strategies: Direct proofs, contradiction, contrapositive, and proof by cases.
  
  
  
  

6. Inequalities

• Objective: To strengthen understanding of inequalities and their applications.


• Topics:
  
  
  
  • Basic Inequalities: AM-GM, Cauchy-Schwarz, and Jensen's Inequality.
  
  
  • Advanced Inequalities: Holder's Inequality, Schur’s Inequality, and Nesbitt's Inequality.
  
  
  
  

Course Delivery Plan

1. Duration

• Recommended Duration: 9-12 months (based on student proficiency)


• Mode: Live online classes, recorded sessions, practice worksheets, and interactive problem-solving sessions.


• Frequency: 2-3 sessions per week (each session 90 minutes to 2 hours long).

2. Weekly Breakdown

• Weeks 1-4: Introduction to Algebra, Combinatorics, and Geometry. Focus on foundational topics like algebraic identities, permutations & combinations, and basic geometry.


• Weeks 5-8: Intermediate topics in Algebra (polynomials, equations), advanced Combinatorics (graph theory, recursion), and introductory Number Theory (divisibility, primes).


• Weeks 9-12: Advanced Geometry (conic sections, geometric inequalities), Number Theory (modular arithmetic, Diophantine equations), and Induction techniques.


• Weeks 13-16: Intensive problem-solving, focusing on past IMO problems, reinforcing all topics, and improving speed and accuracy.


• Weeks 17-20: Advanced Inequalities, complex problem-solving strategies, and optimization of solutions. Final mock tests and review sessions.

3. Key Features

• Live Interactive Classes: Engage with top instructors for real-time doubt resolution, deep dives into complex problems, and feedback.


• Weekly Problem Sets: Curated sets of problems that align with the IMO syllabus, designed to challenge students and enhance their problem-solving abilities.


• Conceptual Videos and Notes: Detailed videos and downloadable PDFs to reinforce classroom teaching with visual explanations.


• Personalized Feedback: After each assignment, students receive detailed feedback on their approach and solutions.


• Advanced Problem Solving Sessions: Weekly problem-solving challenges where students can apply learned techniques under time constraints.


• Mock IMO Exams: Full-length mock tests designed to simulate the actual IMO, allowing students to experience the time pressure and format of the exam.


• Interactive Study Groups: Students will be grouped into small teams to foster collaboration, discussion, and peer-to-peer learning.

4. Resources

• Books and Reference Materials: Access to IMO-recommended books, as well as supplementary study materials.


• Practice Worksheets: Extra sets of problems for practice and self-study.


• Discussion Forums: A dedicated online forum for asking questions, discussing problems, and sharing resources.

5. Assessment and Tracking Progress

• Weekly Quizzes: Short quizzes at the end of each week to track understanding and retention.


• Monthly Assessments: Full-length monthly tests to gauge progress, identify areas of improvement, and adjust the study plan.


• Regular Mock Exams: Simulated IMO exams to prepare students for the pressure and format of the competition.